Applications of Regression Techniques

181 Pages·2019·2.128 MB·English

by  Manoranjan Pal, Premananda Bharati

Manoranjan Pal · Premananda Bharati
Applications
of Regression
Techniques
Applications of Regression Techniques
Manoranjan Pal (cid:129) Premananda Bharati
Applications of Regression
Techniques
123
ManoranjanPal Premananda Bharati
Economic ResearchUnit Biological Anthropology Unit
Indian Statistical Institute Indian Statistical Institute
Kolkata, West Bengal, India Kolkata, West Bengal, India
ISBN978-981-13-9313-6 ISBN978-981-13-9314-3 (eBook)
https://doi.org/10.1007/978-981-13-9314-3
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Preface
Letusconfessattheoutsetthatthisbookcannotbeconsideredasatextbook.This
book is meant for researchers and teachers of different subjects who use statistical
models and techniques. We restrict our attention to techniques of regression anal-
ysis. To get the full benefit of this technique, one needs to apply it very carefully
and prudently. To do this, it is sometimes necessary to formulate an appropriate
regressionmodel.Regressionisaverypowerfultechnique.Manysituationscanbe
formulatedasorconvertedtoaregressionmodel.Themainthingistoidentifysuch
situations.
The book is unique in the sense that none of these works are published in any
journalsorotherbooks.Theseareoriginalresearchworksoftheauthors.Thisbook
is an outcome of our last 10-year research work. However, most of these results
have been presented in different conferences. The readers will encounter many
novel techniques of regression.
This book starts with an exposition of the regression model and gives the idea
behind taking a regression model so that the later chapters can be easily compre-
hended. The volume then comprises formulation and estimation of regression
decomposition models, hidden Markov chain, contribution of regressors by
set-theoretic approach, calorie poverty rate, aggregate growth rate, etc.
Each of these techniques discussed in this book has applications leading to
answers of many unanswered questions. For example, regression decomposition
techniques lead to finding out intra-household gender inequality of consumptions,
intra-household allocation of resources, calorie poverty rate, and adult equivalent
scales.Theassumptionofatrivariatedistributionofthenutrientintake,saycalorie
intake, the income, and the nutrient norm of the households leads us to linear or
log-linearregressionequationsdependingonthetypeofjointdistributionassumed
forthepurposeofestimation.Theset-theoreticapproachtofindthecontributionof
each regressor is another novel technique. Since the explanatory variables are
interrelated, apart from the contribution of individual variation of each of the
explanatory variables, there are joint effects of these variables on the response
variable. We can separate out these effects by the set-theoretic approach. This
conceptcanbeusedtofindaformulawhichshowstheextentofmulticollinearityin
a k-regressor setup. Formulating a hidden Markov chain is again a skilled work
which needs knowledge of the situation in which this formulation is possible. The
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vi Preface
hidden Markov chain model can be transformed into appropriate simultaneous
regressionequationmodel.Ithasmanyapplicationsincludingforecastingtheresult
of the election of subsequent years. Finding aggregate growth rate from individual
growthratethroughgeometricmean(GM)issometimesimpossiblebecauseofthe
fact that there may be a negative component. Since the growth rate is nothing but
ratio, GM is preferable. There is a novel idea of getting aggregate growth rate
through a modification of GM. This technique can be applied even if any of the
growth rates is zero. The modified GM can be found by regression technique after
due modification. The techniques are most general and can be applied in other
similar situations.
Portions of Chaps. 2 and 4 are drawn from the authors’ report of the project
‘Development of Methodology towards Measurement of Poverty’; the project was
sponsored by the Ministry of Statistics and Programme Implementation, Govern-
ment of India. The copyright is, however, not violated. Also in Chap. 9, we bor-
rowed the concept of one of our recent published papers, where the set-theoretic
concept of the relative importance of the regressors was first introduced.
We have tried to describe the techniques in a very simple manner. Even the
difficult concepts are easily understood. This book should be read as a storybook
fromthebeginningtotheend.All thestepsareexplained inasmoothmannerand
often using examples.
Kolkata, India Manoranjan Pal
Premananda Bharati
Contents
1 Introduction to Correlation and Linear Regression Analysis . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Simple Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Simple Linear Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 7
1.4 A Generalization to LS Estimation. . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Comparison of Different Forms . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Multiple Linear Regression Model. . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Transformation of Nonlinear Functions . . . . . . . . . . . . . . . . . . . 14
1.8 The Linear Stochastic Regression Model . . . . . . . . . . . . . . . . . . 14
1.9 Test of Significance for the Sample Correlation Coefficient. . . . . 16
1.10 Testing for Overall Significance of a Regression . . . . . . . . . . . . 16
1.11 Partial Correlation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.12 Rank Correlation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Regression Decomposition Technique Toward Finding
Intra-household Gender Bias of Calorie Consumption . . . . . . . . . . . 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Need for Calorie Decomposition. . . . . . . . . . . . . . . . . . . . . 24
2.3 The Model and the Methodology. . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Glossary and Abbreviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Estimation of Poverty Rates by Calorie Decomposition Method. . . . 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Smoothing of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Calculation of Poverty Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Estimating Calorie Poverty Rates Through Regression. . . . . . . . . . . 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Earlier Attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Objective of the Present Chapter . . . . . . . . . . . . . . . . . . . . . . . . 62
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viii Contents
4.4 The Proposed Calorie Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Calorie Poverty Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 The Poverty Lines and the Poverty Rates. . . . . . . . . . . . . . . . . . 68
4.7 A New Method of Finding Poverty Rate: Error Distribution
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7.1 The Formulation of the Model . . . . . . . . . . . . . . . . . . . 71
4.8 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.9 Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Appendix: The Detail Methodology Toward Finding Poverty Rate . . . . 83
5 Prediction of Voting Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Prediction of Party Positions Assuming Independent
Movement of Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Prediction of Party Positions Assuming Correlated Structure
of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 Finding Aggregate Growth Rate Using Regression Technique . . . . . 105
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.1 Cross-Section Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.2 Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Tackling Zero or Negative Growth Rates. . . . . . . . . . . . . . . . . . 111
6.4 Case of Negative Growth Rates Below Minus 1. . . . . . . . . . . . . 112
6.5 Estimation of Intermediate Growth Rate. . . . . . . . . . . . . . . . . . . 114
6.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7 Testing Linear Restrictions of Parameters in Regression
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 To See Whether a Coefficient is Equal to a Specific Value . . . . . 125
7.3 To See Whether Two Coefficients are Equal . . . . . . . . . . . . . . . 126
7.4 To See Whether a Specific Linear Combination of the
Coefficients is Always Constant. . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5 Applications Using a Set of Artificial Data. . . . . . . . . . . . . . . . . 127
7.5.1 To Test Whether a Coefficient
Takes a Specific Value. . . . . . . . . . . . . . . . . . . . . . . . . 129
7.5.2 To Test Whether a Linear Combination of Parameters
Takes a Specific Value. . . . . . . . . . . . . . . . . . . . . . . . . 130
7.6 Testing Cross-equation Restrictions . . . . . . . . . . . . . . . . . . . . . . 131
7.7 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Contents ix
8 The Regression Models with Dummy Explanatory Variables . . . . . . 135
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2 The Regression Models with Dummy Explanatory Variable . . . . 137
8.3 Polychotomous Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4 Interaction Terms and Their Use . . . . . . . . . . . . . . . . . . . . . . . . 142
8.5 Both Dependent and Explanatory Variables are Qualitative. . . . . 144
8.6 Qualitative and Quantitative Explanatory Variables. . . . . . . . . . . 145
8.6.1 Intercept Dummy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.6.2 Slope Dummy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.6.3 Both Slope and Intercept Dummy . . . . . . . . . . . . . . . . . 146
8.6.4 Both Slope and Intercept Dummy: Discussions . . . . . . . 147
8.7 Asymmetric Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.8 The Chow Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.8.1 The Case Where n < k . . . . . . . . . . . . . . . . . . . . . . . . 149
2
8.8.2 The Use of Dummy Variable to Test the Equality
of Structural Parameters of the Two Periods . . . . . . . . . 149
8.9 Detection of Outlying Observations by Dummy Variables. . . . . . 150
8.10 Prediction by Dummy Variables . . . . . . . . . . . . . . . . . . . . . . . . 151
8.11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9 Relative Contribution of Regressors . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.2 The Concept of Explained and Unexplained Variances
in Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.3 Different Measures of Relative Importance. . . . . . . . . . . . . . . . . 159
9.3.1 Allocation Beta-Squared. . . . . . . . . . . . . . . . . . . . . . . . 159
9.3.2 Allocation First: Squares of Simple Correlations
or Zero-Order Correlations . . . . . . . . . . . . . . . . . . . . . . 160
9.3.3 Allocation Last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.3.4 Hoffman–Pratt Decomposition of R2 . . . . . . . . . . . . . . . 160
9.3.5 LMG or Shapley Decomposition. . . . . . . . . . . . . . . . . . 162
9.3.6 Relative Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.4 The Set-Theoretic Approach to Relative Importance . . . . . . . . . . 164
9.4.1 Degree of Multicollinearity. . . . . . . . . . . . . . . . . . . . . . 165
9.4.2 An Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.5 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
References.... .... .... .... ..... .... .... .... .... .... ..... .... 171
About the Authors
Manoranjan Pal former Professor at the Economic Research Unit (ERU), Indian
Statistical Institute (ISI), obtained his Ph.D. in Statistics from the Indian Statistical
Institute. He has held visiting faculty positions at academic and government
institutions in various countries, including the National Statistical Office of Mon-
golia; South Asia Institute (SAI), University of Heidelberg, Germany; Centre for
Operations Research and Econometrics (CORE), Louvain la Neuve University,
Belgium; Institute of Mathematical Sciences, Faculty of Science, University of
Malay, Malaysia; and the Department of Statistics, Rajshahi University, Bangla-
desh.HewasMemberSecretaryoftheInternationalStatisticalEducationCentrefor
morethantenyears.Hehaspublishedover100researchpapersinreputednational
andinternationaljournalsandbooks.Hisresearchinterestsincludemeasurementof
poverty, inequality and segregation, applied econometrics, measurement of health
and nutrition status, gender bias and empowerment of women.
Premananda Bharati former Professorat theIndianStatistical Institute, obtained
his Ph.D. in Anthropology from the University of Calcutta. He has extensive
teaching experience, including teaching courses in Anthropology to graduate and
postgraduate students at the Indian Statistical Institute and other universities,
includinginstitutesabroad.HeisanexaminerandpapersetterinAnthropologyfor
variousuniversities,andreviewspapersforanumberofjournals.Hehaspublished
more than 100 papers in reputed national and international journals. His research
interestsincludenutritionandhealth,anthropometry,empowermentofwomenand
tribal communities.
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